Number Theory in Progress [Reprint 2012 ed.] 9783110285581, 9783110157154

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Number Theory in Progress

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1 9 9 9

Number Theory in Progress Proceedings of the International Conference on Number Theory organized by the Stefan Banach International Mathematical Center in Honor of the 60th Birthday of Andrzej Schinzel Zakopane, Poland, June 30-July 9, 1997

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Volume 1 Diophantine Problems and Polynomials Editors

Kaiman Györy Henryk Iwaniec Jerzy Urbanowicz

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Walter de Gruyter · Berlin · New York 1999

Editors Κ. Györy Institute of Mathematics and Informatics Lajos Kossuth University 4010 Debrecen, Hungary

H. Iwaniec Department of Mathematics Rutgers University New Brunswick, NJ 08903-2101 USA

J. Urbanowicz Institute of Mathematics Polish Academy of Sciences P.O. Box 137 00-950 Warszawa Poland

1991 Mathematics Subject Classification: 11-02, 11-06, llAxx, HBxx, llCxx, llDxx, llExx, llFxx, llGxx, llHxx, llJxx, llKxx, llLxx, llMxx, llNxx, llPxx, llRxx, llSxx, llTxx, llYxx, 14Gxx, 14Hxx, 14Jxx, 20Bxx, 30Dxx, 32Axx, 33Exx, 40Gxx, 52Cxx, 60Cxx Keywords: aie-conjecture, arithmetic algebraic geometry, automorphic forms, Baker's method, computational number theory, diophantine equations and inequalities, diophantine approximation, elliptic curves, estimates of exponential and character sums, Hecke operators, L-functions and zeta functions, lattices and convex bodies, linear independence, Mahler's measure, Pisot and Salem numbers, polynomials, quadratic forms, sequences and sets, Schinzel's hypothesis, sieves and their applications, spectral theory, transcendental numbers © Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability. Library of Congress —

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International Conference on Number Theory (1997 : Zakopane, Poland) Number theory in progress : proceedings of the International Conference on Number Theory organized by the Stefan Banach International Mathematical Center in honor of the 60th birthday of Andrzej Schinzel, Zakopane, Poland, June 30-July 9, 1997 / editors, Kaiman Györy, Henryk Iwaniec, Jerzy Urbanowicz. p. cm. Contents: v. 1. Diophantine problems and polynomials v. 2. Elementary and analytic number theory. ISBN 3-11-015715-2 (set: acid-free paper) 1. Number theory - Congresses. I. Schinzel, Andrzej. II. Györy, Kälmän. III. Iwaniec, Henryk. IV. Urbanowicz, Jerzy. V. Title QA241.1584 1997 512'.7-dc21 99-19358 CIP

Die Deutsche Bibliothek -

Cataloging-in-Publication-Data

Number theory in progress : proceedings of the International Conference on Number Theory organized by the Stefan Banach International Mathematical Center in honor of the 60th birthday of Andrzej Schinzel, Zakopane, Poland, June 30-July 9, 1997 / ed. Kaiman Györy ... Berlin ; New York : de Gruyter ISBN 3-11-015715-2 Vol. 1. Diophantine problems and polynomials. - 1999

© Copyright 1999 by Walter de Gruyter GmbH & Co. KG, D-10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Typeset using the authors' TEX files: I. Zimmermann, Freiburg. Printing: WB-Druck GmbH & Co., Rieden/Allgäu. Binding: Lüderitz & Bauer, Berlin. Cover design: Thomas Bonnie, Hamburg.

Preface These are the Proceedings of the International Conference on Number Theory held in Zakopane-Koscielisko, Poland, from June 30 to July 9,1997. The conference was organized by the Stefan Banach International Mathematical Center to celebrate the 60th birthday of Professor Andrzej Schinzel. Andrzej Schinzel is the editor-in-chief of Acta Arithmetica—the first international journal devoted exclusively to number theory—for over 28 years. He is also well known for many original results in various areas of number theory appearing in nearly 200 research papers. His main contributions are described in the article of Wladyslaw Narkiewicz in these Proceedings. About 130 mathematicians from 21 countries attended the conference. The focus of the meeting was twofold: Diophantine Problems and Polynomials, and Elementary and Analytic Number Theory. Bogdan Bojarski, Director of the Institute of Mathematics of the Polish Academy of Sciences opened the conference with an address to the participants, and Wladyslaw Narkiewicz delivered the opening lecture on selected works of Andrzej Schinzel. The scientific program was supplemented by a banquet and a one-day excursion, to Cracow, the former capital of Poland. After the excursion, Andrzej Schinzel presented a requested lecture on the history of Poland—a subject which is close to his heart. The Proceedings contain 71 selected, refereed research and survey papers by conference speakers and a few invited mathematicians who were unable to come to the conference. The material is divided into two volumes according to the conference subjects. The articles of the first volume are concerned with diophantine problems and polynomials (diophantine equations, diophantine approximation, transcendental number theory and polynomials). The second volume contains the papers related to elementary and analytic number theory (sieve methods, modular and automorphic forms, Hecke operators, estimates on exponential and character sums, zeta functions and //-functions). A noteworthy feature of these volumes is the large number of papers written by leading mathematicians. Most of the contributions are in English while a few are in French. We thank all the authors and referees for all their contributions to the Proceedings. Many people helped in the organization of the conference or in the editing of the Proceedings. Special thanks are due to Bogdan Bojarski for his guidance and help. We also thank Robert Tijdeman, who played an important role from the initial planning of the conference to the publication of the Proceedings. We thank Lajos Hajdu and Jan K. Kowalski for their efficient secretarial assistance. Special thanks go to Jan K. Kowalski, who looked through the manuscripts, made corrections, and offered valuable suggestions for improving the presentation. The staff of Walter de Gruyter & Co., especially Manfred Karbe, deserve our thanks for an excellent co-operation. We gratefully acknowledge the support of our sponsors: the Stefan Banach Center, the State Committee for Scientific Research of Poland (KBN), the Department of Defense of the Polish government, the Foundation for Polish-German Coopera-

vi

Preface

tion, the Max-Planck-Institut für Mathematik in Bonn, Germany, and the Stefan Batory Foundation. The Organizers want to thank Don Zagier, Director of the Max-Planck-Institut, our first sponsor, for his help and friendly interest. The wonderful blackboards and some electronic equipment, purchased partially from money received from the Max-Planck-Institut, were passed on as a gift to the Mathematical Conference Center in B^dlewo, Poland.

September, 1998

ΚάΙπιάη Györy Henryk Iwaniec Jerzy Urbanowicz

List of participants Scott Ahlgren, Denison University, Granville, USA Francesco Amoroso, University of Torino, Italy Johan Andersson, Stockholm University, Sweden Alan Baker, University of Cambridge, United Kingdom Antal Balog, Mathematical Institute, Hungarian Academy of Sciences, Budapest, Hungary Grzegorz Banaszak, Adam Mickiewicz University, Poznan, Poland Bruce Berndt, University of Illinois at Urbana-Champaign, USA Vasily Bernik, Institute of Mathematics, Academy of Sciences of Belarus, Minsk, Belarus Marie Jose Bertin, Universite Pierre et Marie Curie, Paris, France Frits Beukers, University of Utrecht, The Netherlands Yuri Bilu, ΕΤΗ Zürich, Switzerland Bryan J. Birch, Oxford University, United Kingdom David W. Boyd, University of British Columbia, Vancouver, Canada Bela Brindza, Kossuth Lajos University, Debrecen, Hungary Jerzy Browkin, Warsaw University, Poland W. Dale Brownawell, Penn State University, USA Joerg Brüdern, Universität Stuttgart, Germany Roelof W. Bruggeman, University of Utrecht, The Netherlands V. A. Bykovsky, Russian Academy of Sciences, Far Eastern Branch, Khabarovsk, Russia Mark Coleman, UMIST, Manchester, United Kingdom Jean-Louis Colliot-Thelene, Universite de Paris-Sud, France Giovanni Coppola, DUMA, Universitä degli Studi di Salerno, Italy Daniel Coray, University of Geneva, Switzerland Jean-Marc Couveignes, Universite de Bordeaux, France Alfred Czogala, Silesian University, Katowice, Poland Andrzej D^browski, University of Szczecin, Poland Pierre Debes, Universite Lille, France Hubert Delange, University of Paris 11, France Ulrich Dieter, Technische Universität Graz, Austria Edward Dobrowolski, College of New Caledonia, Prince George, Canada Arturas Dubickas, Vilnius University, Lithuania Ε. H. Dubois, Universite de Basse-Normandie, Caen, France William D. Duke, Rutgers University, New Brunswick, USA

viii

List of participants

Roberto Dvornicich, Universitä di Pisa, Italy Peter D. T. A. Elliott, University of Colorado at Boulder, USA Jan-Hendrik Evertse, University of Leiden, The Netherlands Michael Filaseta, University of South Carolina, USA Kevin Ford, University of Texas at Austin, USA Etienne Fouvry, Universite de Paris-Sud, France Gregory A. Freiman, Tel Aviv University, Israel Michael D. Fried, University of California, Irvine, USA J. B. Friedlander, University of Toronto, Canada Dorian Goldfeld, Columbia University, USA Peter J. Grabner, Technische Universität Graz, Austria George Greaves, University of Wales, Cardiff, United Kingdom Kaiman Györy, Kossuth Lajos University, Debrecen, Hungary Lajos Hajdu, Kossuth Lajos University, Debrecen, Hungary Gabor Haläsz, Mathematical Institute, Hungarian Academy of Sciences, Budapest, Hungary Heini Halberstam, University of Illinois at Urbana-Champaign, USA Franz Halter-Koch, Universität Graz, Austria Adolf Hildebrand, University of Illinois at Urbana-Champaign, USA Jürgen Hurrelbrink, Louisiana State University, USA Martin N. Huxley, University of Wales, Cardiff, United Kingdom Henryk Iwaniec, Rutgers University, New Brunswick, USA Anna Iwaszkiewicz-Rudoszadska, Adam Mickiewicz University, Poznan, Poland Stanislav Jakubec, Institute of Mathematics, Slovak Academy of Sciences, Bratislava, Slovakia Jerzy Kaczorowski, Adam Mickiewicz University, Poznan, Poland Przemyslaw Kaminski, Adam Mickiewicz University, Poznari, Poland Takeshi Kano, Yamagata University, Japan Winfried Kohnen, Universität Heidelberg, Germany Sergei Konyagin, Moscow State University, Russia Katalin Pappne Kovacs, Eötvös University, Budapest, Hungary Emmanuel Kowalski, Rutgers University, New Brunswick, USA Jonas Kubilius, Vilnius University, Lithuania Mieczyslaw Kulas, Adam Mickiewicz University, Poznari, Poland Michel Langevin, Universite de Bordeaux, France Don J. Lewis, National Science Foundation, USA John B. Lewis, Framingham State College, USA Andrzej Minkowski, Warsaw University, Poland Eugenijus Manstavicius, Vilnius University, Lithuania

List of participants

ix

David Masser, University of Basel, Switzerland Michel Mendes Prance, Universite Bordeaux I, France Philippe Michel, Universite de Paris-Sud, Prance Maurice Mignotte, Universite Louis Pasteur, Strasbourg, France Pieter Moree, Max-Planck-Institut für Mathematik, Bonn, Germany Yoichi Motohashi, Nihon University, Japan Wladyslaw Narkiewicz, Wroclaw University, Poland Yu. Nesterenko, Moscow State University, Russia Jean-Louis Nicolas, Universite Claude Bernard, Villeurbanne, France Harald Niederreiter, Austrian Academy of Sciences, Vienna, Austria Werner Georg Nowak, Universität für Bodenkultur, Vienna, Austria Andrew Odlyzko, AT&T Labs - Research, USA Ken Ono, Penn State University, USA Aleksander Pelczyüski, Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland Alberto Perelli, Universitä di Genova, Italy Robert Per lis, Louisiana State University, USA Patrice Philippon, CNRS, Paris, Prance Jänos Pintz, Mathematical Institute, Hungarian Academy of Sciences, Budapest, Hungary Andrew D. Pollington, Brigham Young University, USA, and Imperial College, London, United Kingdom Carl Pomerance, University of Georgia, USA Jacek Pomykala, Warsaw University, Poland Stefan Porubsky, Institute of Chemical Technology, Prague, Czech Republic Maciej Radziejewski, Adam Mickiewicz University, Poznaii, Poland Piotr Rejmenciak, Adam Mickiewicz University, Poznaii, Poland Szilard Revesz, Mathematical Institute, Hungarian Academy of Sciences, Budapest, Hungary Georges Rhin, Universite de Metz, Prance Paulo Ribenboim, Queen's University, Kingston, Ontario, Canada Herman te Riele, Centre for Mathematics and Computer Science (CWI), Amsterdam, The Netherlands Andrzej Rotkiewicz, Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland Michael Rubinstein, Princeton University, USA Zeev Rudnick, Tel Aviv University, Israel Imre Z. Ruzsa, Mathematical Institute, Hungarian Academy of Sciences, Budapest, Hungary Andras Sarközy, Eötvös Lorand University, Budapest, Hungary

χ

List of participants

Andrzej Schinzel, Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland Hans Peter Schlickewei, Universität Marburg, Germany Wolfgang M. Schmidt, University of Colorado, USA Wolfgang Schwarz, University of Frankfurt, Germany Hakim Smati, Universite de Limoges, Prance Chris Smyth, Edinburgh University, United Kingdom Lawrence Somer, Catholic University of America, Washington, USA Vera Sos, Mathematical Institute, Hungarian Academy of Sciences, Budapest, Hungary Cameron L. Stewart, University of Waterloo, Canada Leo Summerer, University of Vienna, Austria Peter Swinnerton-Dyer, Isaac Newton Institute, Cambridge University, United Kingdom Janusz Szmidt, Military University of Technology, Warsaw, Poland Bogdan Szydlo, Adam Mickiewicz University, Poznaii, Poland Kazimierz Szymiczek, Silesian University, Katowice, Poland Jörg Maximilian Thuswaldner, Montanuniversität Leoben, Austria Robert F. Tichy, Technische Universität Graz, Austria Robert Tijdeman, University of Leiden, The Netherlands Gerhard Turnwald, University of Tübingen, Germany Yoichi Uetake, Adam Mickiewicz University, Poznaii, Poland Jerzy Urbanowicz, Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland Jeffrey D. Vaaler, The University of Texas at Austin, USA Carlo Viola, Universitä di Pisa, Italy Michel Waldschmidt, Universite Pierre et Marie Curie, Paris, France Gary Walsh, University of Ottawa, Canada Eduard Wirsing, Universität Ulm, Germany Kunrui Yu, Hong Kong University of Science and Technology, Hong Kong / China Don Zagier, Max-Planck-Institut für Mathematik, Bonn, Germany Umberto Zannier, Istituto Universitario di Architettura, Venezia, Italy Tao Zhan, Shandong University, Jinan, China

Table of contents of Volume I Preface

ν

List of participants

vii

Table of contents of Volume I

xi

Table of contents of Volume II

xv

Quelques nouveaux resultats sur les nombres de Pisot et de Salem M.J. Bertin

1

Irreducibility of polynomials and arithmetic progressions with equal products of terms F. Beukers, T.N. Shorey and R. Tijdeman

11

Mahler's measure and special values of L-functions — some conjectures David W. Boyd

27

On the distribution of solutions of Thue's equation Bela Brindza, A kos Pinter, Alfred J. van der Poorten and Michel Waldschmidt

35

Linear independence and divided derivatives of a Drinfeld module. I W. Dale Brownawell

47

Cubic threefolds with six double points D.F. Coray, D.J. Lewis, N.I. Shepherd-Barron Dyer

63

and Sir Peter

Swinnerton-

Arithmetique et espaces de modules de revetements Pierre Debes

75

On a polynomial with large number of irreducible factors A. Dubickas

103

Fractions continues parametrees et critere de Rabinowitsch E. Dubois et A. Farhane

111

The Absolute Subspace Theorem and linear equations with unknowns from a multiplicative group Jan-Hendrik Evertse and Hans Peter Schlickewei

121

On the factorization of polynomials with small Euclidean norm Michael Filaseta

143

Small Salem numbers V. Flammang, M. Grandcolas and G. Rhin

165

Variables separated polynomials, the genus 0 problem and moduli spaces Michael D. Fried

169

xii

Table of contents of Volume I

Some polynomial identities related to the ai>c-conjecture George Greaves and Abderrahmane Nitaj

229

On the distribution of solutions of decomposable form equations K. Gyory

237

Finding small degree factors of lacunary polynomials H. W. Lenstra, Jr.

267

On the factorization of lacunary polynomials H. W. Lenstra, Jr.

277

Specializations of some hyperelliptic Jacobians D. W. Masser

293

Salem numbers Pisot numbers stars J.F. McKee, P. and Rowlinson and C.J.from Smyth On lacunary formal series and their continued fraction expansion Michel Mendes France, Alfred J. van der Poorten and Jeffrey Shallit

309 321

2

The ultra-divergent series Σ „ > 0 0~ " M. Mendes France and A. Sebbar

327

Une remarque sur l'equation de Catalan Maurice Mignotte

337

The work ofNarkiewicz Andrzej Schinzel in number theory Wladyslaw

341

Algebraic curves with many rational points over finite fields of characteristic 2 Harald Niederreiter and Chaoping Xing 359 Arf equivalence I Robert Perils

381

The number of irreducible factors of a polynomial, III Christopher G. Pinner and Jeffrey D. Vaaler

395

Identities with covering systems and Appell polynomials Stefan Porubsky

407

Binary recurring sequences and powers, I Paulo Ribenboim

419

On Mahler's measure for polynomials in several variables Imre Z. Ruzsa

431

Polynomials that divide and many fc-nomials Hans Peter Schlickewei Viola modulo prime powers Solution trees of polynomial Carlo congruences Wolfgang M. Schmidt

445 451

Table of contents of Volume I The equation α

xiii

^ = byq with ab > 1

T.N. Shorey

473

Transcendence bases of the algebra of vector invariants for a symmetric group Serguei A. Stepanov 487 Some applications of Schinzel's Hypothesis to Diophantine equations Sir Peter Swinnerton-Dyer

503

On the Milnor exact sequence for rational quadratic forms Kazimierz Szymiczek

531

Some notes on monodromy groups of polynomials Gerhard Turnwald

539

Integer valued entire functions on Cartesian products Michel Waldschmidt

553

On a conjecture of Schinzel and Tijdeman P.G. Walsh

577

List of contributors

583

Table of contents of Volume II Table of contents of Volume II

ν

On some convex lattice polytopes Antal Balog and Jean-Marc Deshouillers

591

Digital blocks in linear numeration systems G. Barat, R.F. Tichy and R. Tijdeman

607

On irregularities of distribution in shifts and dilation of integer sequences, II J. Beck, A. Särközy and C.L. Stewart 633 Addition of integer sequences and subsets of real tori Yuri Bilu

639

Kloosterman sums for the modular group Roelof W. Bruggeman

651

Hecke series values of holomorphic cusp forms in the centre of the critical strip V.A. Bykovsky 675 Bounds for frequencies of residues of regular second-order recurrences modulo pr Walter Carlip and Lawrence Somer

691

Differential inequalities for Iwaniec's q functions Harold G. Diamond and H. Halberstam

721

When is the product of two Hecke eigenforms an eigenform? W. Duke

737

Grandes valeurs de la fonction dk Jean-Luc Duras, Jean-Louis Nicolas et Guy Robin

743

On a multiplicative analogue of Goldbach's conjecture P.D.T.A. Elliott

771

On two conjectures of Sierpinski concerning the arithmetic functions σ and φ Kevin Ford and Sergei Konyagin 795 Residue classes free of values of Euler's function Kevin Ford, Sergei Konyagin and Carl Pomerance

805

Gauss' congruence from Dirichlet's class number formula and generalizations Glenn J. Fox, Jerzy Urbanowicz and Kenneth S. Williams 813 Note on a variance in the distribution of primes J.B. Friedlander and D.A. Goldston

841

The distribution of modular symbols Donan Goldfeld

849

xvi

Table of contents of Volume II

On the solutions to φ(η) = φ(τι + k) S. W. Graham, Jeffrey J. Holt and Carl Pomerance

867

Lattice points in the sphere D.R. Heath-Brown

883

On the Barban-Davenport-Halberstam theorem: XII C. Hooley

893

The integer points close to a curve III M.N. Huxley

911

Dirichlet L-functions at the central point H. Iwaniec and P. Sarnak

941

The Selberg class: survey J. Kaczorowski anda A. Perelli A radically simplified Selberg zeta function for the modular group John B. Lewis

953 993

The Goldbach-Vinogradov Theorem Jianya Liu and Tao Zhan

1005

Α Tauber theorem and multiplicative functions on permutations Eugenijus Manstavicius

1025

Extreme values of Dirichlet L-functions at 1 H.L. Montgomery

and R.C.

Vaughan

1039

On the remainder term in the Selberg sieve Yoichi Motohashi

1053

Newforms for the modular group on spaces of dimension 2 R.A. Rankin

1065

Computational sieving applied to some classical number-theoretic problems Herman te Riele 1071 Evaluation of mean-values of products of shifted arithmetical functions, II Wolfgang Schwarz

1081

Crible d'Eratosthene et modele de Kubilius Gerald Tenenbaum

1099

Three two-dimensional Weyl steps in the circle problem. III. Exponential integrals and application Ulrike Μ.Α.

Vorhauer and Eduard Wirsing

1131

From quadratic functions to modular functions D. Zagier

1147

List of contributors

1179

Quelques nouveaux resultats sur les nombres de Pisot et de Salem M.J. Bertin

Au Professeur Schinzel, pour son soixantieme

anniversaire

Resume. Apres un court historique sur le probleme de Lehmer, nous montrons l'importance de la repartition des conjugues d'un entier algebrique pour revaluation de sa mesure de Mahler.

Les nombres de Pisot et de Salem apparaissent dejä dans le celebre article de D.H. Lehmer (1933) [Le], Motive par la recherche de grands nombres premiers ä partir des quantites r

Δ „ ( Ρ ) = Π (α™ — 1) oü Ρ est un polynöme unitaire, irreductible, ä coefficients eni—1 tiers rationnels possedant les racines c*i,..., a r , Lehmer observa que pour obtenir sans trop de calculs de tres grands nombres premiers, les quantites ne devaient pas croitre trop vite. Or si Ρ ne possede pas de racine de module 1, on deduit aisement que lim η—>oo

»n+1 (Ρ) Δη(Ρ)

r

= IJmax(|a i |,l) = M(P), i=1

oil M(P) designe la mesure de Mahler du polynome unitaire P. Lehmer s'interessa d'abord aux mesures de Mahler des polynömes unitaires irreductibles non reciproques. II montra que: en en en en

degre degre degre degre

1, 2, 3, 4,

celui celui celui celui

de de de de

plus plus plus plus

petite petite petite petite

mesure mesure mesure mesure

est est est est

X — 2 de mesure 2; X 2 — X — 1 de mesure 1+2v/^; X 3 — X — 1 de mesure 1,3247...; X 4 — X — 1 de mesure 1,38027...

Faute d'outils de calcul et parce que les plus petites mesures obtenues jusqu'au degre 4 etaient obtenues avec des trinömes, Lehmer se contenta d'etudier les trinömes en degre 5, 6 et 7. II obtint pour plus petites mesures de Mahler: en degre 5, le polynöme X 5 — X 3 — 1 de mesure 1,3625...;

M.J. Bertin

2

en degre 6, le polynome X 6 — X — 1 de mesure 1,3707...; en degre 7, le polynome X 7 — X 3 — 1 de mesure 1,3797... Observons que jusqu'au degre 4 les polynömes cites ne possedent qu'une seule racine θ, θ > 1, exterieure au disque unite. Cette racine θ est un nombre de Pisot (on trouve aussi dans la litterature P - V nombre, Ρ pour Pisot et V pour Vijayaraghavan [V] qui etudia egalement ces nombres). Un nombre de Pisot est done un entier algebrique θ, θ > 1, dont tous les autres conjugues ont un module strictement inferieur ά 1. L 'ensemble des nombres de Pisot est note S. Observons egalement que vers les annees 1960, les resultats suivants furent obtenus: 2 = inf S",

1 +

^

= inf 5",

6>o = 1,3247... = inf S.

0i = 1,38027... est le deuxieme plus petit nombre de Pisot [G], [D-P1,2,3]. (L'ensemble S' designe l'ensemble derive de l'ensemble S). Lehmer observa encore que les polynömes reciproques non cyclotomiques de plus petite mesure sont ceux qui possedent toutes leurs racines sauf 2 sur le cercle unite. Ces entiers algebriques deviendront vite celebres: ce sont les nombres de Salem. Leur ensemble est note T. Un nombre de Salem est done un entier algebrique τ, τ > 1, dont tous les autres conjugues ont un module inferieur ou egal a 1, avec effectivement des conjugues de module 1. Lehmer donna les plus petites mesures pour les polynömes reciproques par degre croissant: X2 X4 X6 X8

-

3X X3 X4 X5

+ -

1 de mesure 2,618 . . . ; X 2 - X + 1 de mesure 1,7220...; X 3 - X 2 + 1 de mesure 1,4012 . . . ; X 4 - X 3 + 1 de mesure 1,2806...

Ces trois dernieres mesures sont les plus petits nombres de Salem de degre 4,6,8 respectivement. On le verifie aisement dans les tables de Boyd [Bo 2] (1980). En degre 10, Lehmer avoua n'avoir pu tout examiner mais donna sa meilleure mesure obtenue, 1,1762... Ce nombre est l'unique racine de module superieur ä 1 du polynome: X 1 0 + X 9 - X 7 - X 6 - X 5 - X 4 - X 3 + X + 1. Ce nombre est appele aujourd'hui nombre de Lehmer. Alors, tout naturellement, Lehmer posa la question suivante: existe-t-il des polynömes unitaires, irreductibles, a coefficients entiers de mesure inferieure a 1,1762...?

Quelques nouveaux resultats sur les nombres de Pisot et de Salem

3

C'est la fameuse question de Lehmer que certains auteurs tournent parfois en conjecture de Lehmer ou generalisent ä diverses situations. En depit de toutes les investigations de nombreux mathematiciens, ce nombre demeure toujours le plus petit nombre de Salem connu et sa mesure, qui est le nombre lui-meme, la plus petite mesure de Mahler actuellement connue. Remarquons maintenant la repartition des conjugues dans C du plus petit nombre de Pisot θο = 1,32... ainsi que du plus petit nombre de Salem connu r 0 = 1,1762... Le nombre 0Q a deux conjugues de module —ί=- et d'arguments ±139°67. Le nombre To possede un conjugue et ses autres conjugues de module 1 et d'arguments ±62°8, ±106°9, ±137°2, ±160°6. Pourquoi regarder aussi attentivement les conjugues de ces nombres? Au debut de leur etude, lors de la these de Pisot [P] et dans les quelques 30 ans qui suivirent, on s'interessait essentiellement au nombre lui-meme, c'est-ä-dire aux proprietes du plus grand des conjugues 0 ou r . II y a d'ailleurs des resultats celebres: S est un ensemble ferme pour la topologie de Μ (Salem [S1] (1944)); S est contenu dans l'ensemble derive Τ' des nombres de Salem (Salem [S2] (1945)); M(a) > ÖQ = 1,32... = inf S st a est un entier algebrique non reciproque (Smyth [Sm] (1970)); tout nombre de Salem peut etre obtenu ä partir d'un nombre de Pisot par la construction de Salem (Boyd [Bol] (1977)). Cependant, si l'on s'interesse ä la question de Lehmer, la repartition des conjugues est tres importante. J'en veux pour preuve deux resultats. Le premier est un resultat de Zagier (1993) [Za]: Tout nombre algebrique a, de degre d, α φ 0, α φ 1, α φ (1 + l'inegalite:

3)/2, verifie

Une telle formule renseigne sur la localisation des zeros des entiers algebriques de petite mesure. Par exemple, on deduit du resultat de Zagier qu'un nombre de Pisot a de degre 3, de mesure inferieure ä 2, a ses conjugues de module inferieur ä 1 ä l'exterieur du cercle de centre 1 et de rayon 1. Parmi ces nombres, il y a le plus petit nombre de Pisot θο = 1,32... racine de l'equation X 3 — X — 1 = 0 ainsi que le quatrieme plus petit nombre de Pisot θ = 1,46557... racine de l'equation X 3 — X 2 — 1 dont les autres conjugues ont pour module ^ et argument ±106°, 36. On deduit egalement du resultat de Zagier qu'il n'existe pas de nombre de Salem τ < 2 dont les conjugues de module 1 aient un argument < | en module. En outre, pour τ > 2, on a τ > C(d), la constante C(d) dependant du degre d de τ et tendant vers l'infini lorsque d tend vers l'infini.

4

M.J. Bertin

Ces exemples montrent que la repartition des conjugues et done la trace et la norme de l'entier algebrique ont une influence sur la mesure. C'est le deuxieme resultat evoque precedemment, le resultat de Matveev, qui m'a permis de concretiser cette remarque. Citons d'abord le resultat de Matveev [M]. Soit Κ un corps de nombres de degre d. Notons les corps conjugues de Κ sur C. II leur correspond les valeurs absolues archimediennes normalisees, \α\σ = la1·"11, 1 < σ < d, a £ Κ*, tous les plongements etant consideres, bien que deux plongements complexes conjugues definissent la meme valeur absolue. Designons les ideaux premiers de Κ avec des indices σ, σ > d. II leur correspond des valeurs absolues non archimediennes \α\σ = N(V)~m, V = Τσ, σ > d,m etant l'exposant de V dans la decomposition en ideaux premiers de l'ideal (a). Nous avons alors pour α Φ 0 la formule du produit Π σ \α\σ = 1, oü seulement un nombre fini de facteurs differe de 1. Notons T> = T>{K) 1'ensemble { 1 , 2 , . . . , d} et choisissons un sous-ensemble S, S C T> et pour tout σ £ S, une determination de Definissons μ = μ(α,δ)

= (1/2) £ | l n |α| σ |,

Λ = λ ( a , S ) - (1/5) £ | ΐ η α < " > a€S

oil S = Card (S). Theoreme 0. Avec les notations precedentes, on a, pour α φ 1, l'inegalite sh(A/2) > exp(-(dln2 + μ)/5). Corollaire 0. Avec les notations du theoreme, il en resulte l'inegalite Λ > (21n/3)exp(—(iiln2 + μ)/S), oil β = (3 + \/5)/2 est le carre du nombre d'or. Avant de donner nos resultats, rappelons egalement la celebre inegalite de Schinzel [Sc] (1973):

si a est un entier algebrique totalement reel. Theoreme 1 [Be 3]. Soit θ un entier algebrique totalement reel de degre d, de norme Ν (θ). Si θ est totalement positif, on a: Μ(θ) > Ma Sinon Μ(θ) > Max

Quelques nouveaux resultats sur les nombres de Pisot et de Salem

5

Nous allons maintenant etudier une classe particuliere d'entiers algebriques. Designons par Θν l'ensemble des entiers algebriques reciproques θ ayant au moins un conjugue de module 1 et les 2v conjugues de module different de 1 reels. Le theoreme suivant montre que certaines classes de nombres de Θυ se comportent presque comme des entiers algebriques reciproques totalement reels. Theoreme 2. On considere l'ensemble des θ G θν de degre 2d tels que d/v tende vers 7 > 1 lorsque d tend vers +oo. Sott Xq la racine superieure ά 1 de l'equation X 2 - ( 2 + 2 - 2 ( 7 " 1 ) ) X + l = 0. Alors, pour tout ε > 0, il existe un entier do tel que pour tout d, d > do, I'on ait Inf

Μ{θ) >

deg θ—d

Si 7=1,

Χο(1~ε)/2.

on retrouve asymptotiquement le resultat de Schinzel.

Preuve. On peut supposer θ Ε θ„ totalement positif. Considerons Κ = Q(9) et prenons pour S l'ensemble des valeurs absolues archimediennes de Κ sauf Celles correspondant aux conjugues de module 1. On a done S = 2u, μ = 0, Λ = {I/ν) InΜ{θ). La fonction (Ι/λ) sh(Ä/2) etant decroissante, si λ < In Xq, alors (1/Λ) sh(A/2) < 2 - 7 / l n X o > d'oü l'inegalite.

•

Les nombres de Salem sont tres loin de verifier la condition du theoreme 2. Cependant, on peut deduire pour eux un resultat sur la repartition sur le cercle unite de leurs conjugues de module 1. Theoreme 3. Soit τ un nombre de Salem de degre 2d. Soient ,..., θ^-ι les arguments des conjugues de module 1 de τ compris entre 0 et π. Alors si β = (3 + λ/5)/2, on α l'inegalite suivante 1 —

g

g i

™ ( 2d In 2 + In r λ >(2M)exp()• 2(d_1}

En particulier, si τ appartient a l'ensemble des nombres de Salem inferieurs ά 1,3 on a pour d assez grand, 1

d-l i—l

Preuve. Si Κ = Q(r), S = V-{l,a} ού σ(τ) = Ι / τ , alors μ = Ιητ, S = 2(d- 1), Λ= ΣΊζΙ θί. L'inegalite resulte alors du corollaire 0. • Remarque. Ce type de resultats est ä rapprocher des resultats de Bertin-Boyd [B-B 1,2].

6

M.J. Bertin

Nous pouvons maintenant expliquer pourquoi les conjugues exterieurs au cercle unite des entiers algebriques reciproques de petite mesure sont autant que possible imaginaires. Definition. On appelle f-Salem un entier algebrique reciproque ayant u conjugues exterieur au cercle unite et au moins un conjugue de module 1. Nous dirons que le i/-Salem est totalement reel si ses conjugues exterieurs au cercle unite sont tous reels. On deduit alors directement du corollaire 0 le theoreme suivant. Theoreme 4. La mesure de Mahler d'un v-Salem θ de degre 2i/ + 2k, totalement reel, verifie I'inegalite Μ{θ) > ((1 + λ/5)/2γ' 2 "'" =

C{v,k).

Les v-Salem totalement positifs, en particulier les nombres de Salem qui sont des 1-Salem, verifient Μ(θ) > (C(u,k))2. Corollaire. (1) Un nombre de Salem de degre 4 est superieur ά 1,618033... Le plus petit d'entre eux est donne par la table de Boyd et vaut 1,722683... (2) Un 2-Salem de degre 6, de mesure inferieure a 1,9749... est totalement imaginaire. Le seul 2-Salem de mesure inferieure a 2 est de mesure 1,9962... et est encore totalement imaginaire. (3) Un 2-Salem de degre 8, de mesure inferieure a 1,618033... est totalement imaginaire et tous ceux de mesure inferieure ά 2 sont ou bien totalement imaginaires ou bien ont leurs conjugues exterieurs au cercle unite de signes contraires. Le seul 2-Salem de degre 8 de mesure 1,8475... inferieure ά 2 possede deux conjugues exterieurs au cercle unite reels et de signes contraires. Tous les autres de mesure inferieure a 2 sont totalement imaginaires. (4) Un 2-Salem de degre 10, de mesure inferieure a 1,4053... est totalement imaginaire, de mesure inferieure ä 1,9749... est soit totalement imaginaire soit possede ses deux conjugues exterieurs au cercle unite de signes contraires. Les seuls 2-Salem de mesure inferieure ά 2 ayant deux conjugues exterieurs au cercle unite reels de signes contraires sont les 2-Salem de mesure 1,835053.. 1,836868..., 1,953585..1,961647 ...et 1,994976... Les 3-Salem et les 4-Salem de degre 10 de mesure inferieure ä 2 ont au moins deux conjugues exterieurs au cercle unite imaginaires conjugues. Preuve. Ces resultats decoulent immediatement du theoreme 4 et du calcul des conjugues des entiers reciproques de petite mesure donnes par la table de Boyd [Bo 2]. Nous laissons au lecteur le soin de completer ce corollaire en degre superieur grace ä [Bo 3], [Bo 4], •

Quelques nouveaux resultats sur les nombres de Pisot et de Salem

7

J'en viens maintenant aux resultats de mes eleves: ceux de Zaimi (1994-1997) concernent plus particulierement les nombres de Pisot et ceux de Lalande (1996-1997) les nombres de Salem. Le travail de Za'imi porte sur une generalisation des nombres de Pisot. Etant donne un corps de nombres Κ et un entier algebrique θ, θ > 1, de polynöme minimal Ρ sur K, on dit que θ est un X-nombre de Pisot ou encore appartient ä S k si pour tout plongement de Κ dans C, σ(Ρκ) possede une unique racine de module superieur ä 1 et aucune racine de module 1. La definition est due ä A.M. Berge et J. Martinet [Β-Μ], J'avais remarque que si Κ est un corps quadratique reel, les elements de Sk de mesure inferieure ä 2,6 sont en nombre fini et leur mesure croit avec le discriminant du corps de nombres [Be 2]. Za'imi a continue l'etude pour les corps quadratiques imaginaires et les corps cubiques totalement reels. II a prouve dans sa these (1994) [Ζ 1] que la mesure d'un ii-nombre de Pisot θ verifie l'inegalite:

si Κ est quadratique, et MOT >

^

si Κ est cubique totalement reel, Δ designant le discriminant de Κ. II a ensuite prouve [Z 2] le theoreme suivant: Soient Κ un corps de nombres totalement reel primitif de degre d ou bien un corps quadratique de discriminant D et Ρ le polynöme minimal sur Κ d 'un entier algebrique Θ. Si le polynöme minimal de θ sur Κ est non reciproque et si pour tout plongement σ de Κ dans C les polynömes Ρ et σΡ sont premiers entre eux, alors > \D\_ M{9)2(d-1) dr Une question demeurait cependant ouverte depuis 1989 [B-M]. Plongeons Sk dans l'algebre A = R r i χ C2 par la suite (θσ)σ des conjugues de θ de module superieur ä 1, (ri,Γ2) designant la signature de Κ. L 'ensemble Sk est-il ferme dans ΑΊ Lorsque Sk est ferme, quels sont les elements de mesure minimale? Zaimi vient de demontrer (1997) [Z3] l'equivalence suivante: SK EST ferme dans A si et seulement si Κ = Q ou Κ quadratique imaginaire. (On savait dejä, grace ä Salem [S1] pour Κ = Q et ä Zaimi [Ζ 1] pour Κ quadratique imaginaire, que Sk est ferme.) Zaimi a egalement prouve [Z 3], que si Κ est quadratique imaginaire ou totalement reel, alors: infSW = 6>o = 1,32...

8

M.J. Bertin

et =

—

-

—

.

La preuve utilise encore le resultat de Schinzel precedemment cite ainsi que la remarquable propriete des nombres de Pisot: Tout corps de nombres reel peut etre engendre par un nombre de Pisot qui est une unite du corps. On peut alors se demander si les corps de nombres imaginaires ayant seulement deux corps conjugues reels possede une pareille propriete, ä savoir d'etre engendre par un nombre de Salem. Lalande (1996-1997) [La] a caracterise ά l'aide de leur groupe de Galois les corps de nombres engendres par un nombre de Salem et plus generalement les corps de nombres non totalement imaginaires engendres par une unite reciproque.

References [B-M]

Berge, A.M., Martinet, J., Notions relatives de regulateurs et de hauteurs. Acta Arith. 54 (1989), 155-170.

[Bei]

Bertin, M.J., if-nombres de Pisot et de Salem. Acta Arith. 68 (1994), 113-131.

[Be2]

— AT-nombres de Pisot et de Salem. In: Advances in number theory (ed. par F.Q. Gouvea et N. Yui), 391-397. Oxford University Press, New York 1993.

[Be 3]

— The operator χ + (Ι/a;) — 2 and the reciprocal integers. In: Proceedings of the 5th Conference of the Canadian Number Theory Association, Ottawa 1996, ä paraitre.

[B-Bl] Bertin, M.J., Boyd, D.W., Une caracterisation de certaines classes de nombres de Salem. C. R. Acad. Sei. Paris Ser. I Math. 303 (1986), 837-839. [B-B 2] — A characterization of two related classes of Salem numbers. J. Number Theory 50 (1995), 309-317. [Bo 1]

Boyd, D.W., Small Salem numbers. Duke Math. J. 44 (1977), 315-327.

[Bo 2]

— Reciprocal polynomials having small measure. Math. Comp. 35 (1980), 1361-1377.

[Bo3]

— Reciprocal polynomials having small measure II. Math. Comp. 53 (1989), 355-357, S1-S5.

[Bo 4]

— Small measures of reciprocal polynomials. Correspondance privee, Avril 1995.

[D-P1] Dufresnoy, J., Pisot, Ch., Sur les petits elements d'un ensemble remarquable d'entiers algebriques. C. R. Acad. Sei. Paris 238 (1954), 1551-1553. [D-P 2] — Etude de certaines fonctions meromorphes bornees sur le cercle unite. Application ä un ensemble ferme d'entiers algebriques. Ann. Sei. Ecole Norm. Sup. (3) 72 (1955), 69-92. [D-P 3] — Sur les elements d'accumulation d'un ensemble ferme d'entiers algebriques. Bull. Sei. Math. (2) 79 (1955), 54-64.

Quelques nouveaux resultats sur les nombres de Pisot et de Salem

9

Grandet-Hugot, M., Ensembles fermes d'entiers algebriques. Ann. Sei. Ecole Norm. Sup. (3) 82 (1965), 1-35. Lalande, F., Corps de nombres engendres par un nombre de Salem. Expose au Seminaire d'Arithmetique, 1996, non publie. Lehmer, D.H., Factorization of certain cyclotomic functions. Ann. of Math. (2) 34 (1933), 461-479. Matveev, Ε.Μ., On algebraic numbers of small logarithmic height. Mat. Zametki, ä paraitre. Pisot, Ch., La repartition modulo 1 et les nombres algebriques. Ann. Scuola Norm. Sup. Pisa (2) 7 (1938), 205-248. Salem, R., A remarkable class of algebraic integers. Proof of a conjecture of Vijayaraghavan. Duke Math. J. 11 (1944), 103-107. — Power series with integral coefficients. Duke Math. J. 12 (1945), 153-171. Schinzel, Α., On the product of the conjugates outside of the unit circle of an algebraic number. Acta Arith. 24 (1973), 385-399. Addendum: ibid. 26 (1974/75), 329-331. Smyth, C.J., On the product of the conjugates outside the unit circle of an algebraic integer. Bull. London Math. Soc. 3 (1971), 169-175. Vijayaraghavan, T., On fractional part of the powers of a number: I, J. London Math. Soc. 15 (1940), 159-160. II, Proc. Cambridge Philos. Soc. 37 (1941), 349-357. III, J. London Math. Soc. 17 (1942), 137-138. IV, J. Indian Math. Soc. (N.S.) 12 (1948), 33-39. Zagier, D., Algebraic numbers close to both 0 and 1. Math. Comp. 61 (1993), 485-491. Zaimi, T., Sur les nombres de Pisot relatifs. These de Doctorat de l'Universite Paris 6, Mai 1994. — Sur les i-f-nombres de Pisot de petite mesure. Acta Arith. 77 (1996), 103-131. — Sur la fermeture de l'ensemble des if-nombres de Pisot. Acta Arith. 83 (1998), 363-367.

Irreducibility of polynomials and arithmetic progressions with equal products of terms F. Beukers, T.N. Shorey and R.

Tijdeman

To Andrzej Schinzel on the occasion of his sixtieth

birthday

Abstract. In some fundamental papers Davenport, Lewis and Schinzel [DLS], Schinzel [Sehl, Sch3] and Pried [Frl, Fr2, Fr3] have shown how irreducibility criteria for polynomials f ( X ) — g(Y) in combination with results of Runge or Siegel can be used to prove the finiteness of the solutions of the corresponding diophantine equation / ( x ) = g(y) in integers x, y. In the present paper we are particularly interested in the case f ( X ) = X(X + di) • • • (X + (m-l)di), g{Y) = Y(Y + d2) • • • (Y + (n-\)d2), i.e. the diophantine equation x{x + dx) • • • (x + (m - l)di) = y(y + d2) • • • (y + (η - 1 )d2). (0.1) We first give some history on this equation and indicate how results for this equation can be derived from general irreducibility theory in the literature. Then we give direct proofs of the results using only basic facts on algebraic curves. 1991 Mathematics Subject Classification: 11D57.

1. W h e n do finite arithmetic sequences have equal products of terms? The question, under the restriction t h a t the arithmetic progressions have equal lengths, was posed in Poland by Gabovich in 1966 [Ga]. He mentioned the example 2 - 6 - 1 0 = 4 - 5 - 6 and gave an infinite class of examples of length 4 including 7 · 20 · 33 • 46 = 20 • 21 · 22 · 23 and 18 • 37 · 56 · 75 = 24 · 37 · 50 · 63. Some infinite classes of solutions of length 5 were given by Szymiczek [Sz] and Choudhry [Ch]. In 1968 Minkowski [MqJ observed that for every positive integer m 2 · 6 · 10 · · · (4m - 2) = (m + 1 )(m + 2) · · · (2m). An opposite result was obtained by Saradha, Shorey and Tijdeman [SaSTl].

(1.1)

12

F. Beukers, T.N. Shorey and R. Tijdeman

T h e o r e m A . For fixed integers d\ > d2 > 0 there are only finitely many integers m > 2, x,y gcd(x, y,d\,d2) = 1 and x(x + d1)---(x+(m-

positive

l)di) = y(y + d2) • • • (y + (m - 1 )d2)

except for the solutions (1.1). The other solutions are effectively

(1.2)

computable.

The special case of equal products of arithmetic progressions of different lengths, but both with difference 1, was also considered by some authors. This is the same as equality of the products of two blocks of positive integers x(x + 1) · · · (x + m — 1) = y(y + 1) · · · (y + η - 1),

m < η.

(1.3)

In 1963 Mordell [Mo] showed that (x, y) = (2,1) and (14,5) are the only solutions of (1.3) in case (m, n) = (2,3). MacLeod and Barrodale [MaB] proved that there are no solutions for (m, n) = (2,4), (2, 6), (2,8), (2,12), (4, 8) and (5,10) and only one solution (x, y) = (8,1) if (m, n) = (3,6). Boyd and Kisilevsky [BoK] found that there are just three solutions when (τη, η) = (3,4), namely (ζ, y) = (2,1), (4,2) and the remarkable (55,19). Other known solutions occur when (m, n) = (3,5), namely (x, y) = (4,1), (8,2), and when (m, η) = (4,7) we find (x, y) = (7,1), (63,8). In the years 1990-1996 Shorey and coworkers studied the equation (0.1) subject to η > τη > 1 with d\, d2 and η/τη fixed. Saradha and Shorey [SaSl] showed that (τη, η, χ, y) = (3,6,8,1) is the only solution of (0.1) with d\ = d2 = 1 and η = 2m. They proved in [SaS2] that (0.1) has no solutions with d\ = d2 = 1 and η = 3τη or η = 4m,. Mignotte and Shorey [MiS] proved that there are no solutions with d\ = d2 = 1 and η = 5m or η = 6πι. The case d\ = 1, d2 > 1, η = τη or η = 2m was considered by Saradha, Shorey and Tijdeman [SaST3]. In the paper all the solutions of equation (0.1) with η = m, d\ = 1, d2 = 2 , 3 , 5 , 6 , 7 , 9 , 1 0 and equation (0.1) with η = 2m, d\ = 1, d2 = 5,6 have been given. In [SaST4] extensions to more general equations on the products of values of a polynomial at points in arithmetic progressions have been treated. Some general finiteness results were obtained by Saradha and Shorey [SaS3] and Saradha, Shorey and Tijdeman [SaST2]. They proved that if η > τη and gcd(m,n) > 1, then the positive solutions m,n,x,y of equation (0.1) can be effectively bounded in terms of d\, d2 and η/τη with the exception of the infinite class of solutions m = 2, η = 4, d\ = 2 d2, χ = y2 + 3d2y. They further proved that if m = 2 or 4 and η > 2 is odd, then the solutions n, x, y of (0.1) can be effectively bounded in terms of d\ and d2. For the remaining case gcd(m, η) = 1 no general effective method is available. In that case, with m, η fixed, we have to resort to Siegel's famous result on integral points on algebraic curves, which is, unfortunately, ineffective. In addition, we can use Faltings's work on Mordell's conjecture to make a similar statement for rational solutions as well. We summarize this in the following theorem, whose proof is the main goal of this paper.

Irreducibility of polynomials and arithmetic progressions

13

T h e o r e m 1.1. Let m and η be integers with 1 < m < n. Let d\ and d2 be positive rational numbers with di ^ d2, if m = n. The equation x(x + di)---(x+(m

- l)di) = y(y + d2) • · · (y + (η - 1 )d2)

admits only finitely many integral solutions x, y except for solutions χ = y2 + 3 d 2 y , —2d2 — 3d2y — y2 when πι = 2, η = over, the equation admits infinitely many rational solutions (2,2), (2,3), (2,4), (3,3) and m = 2, η = 6, d\ = 15^2/4. In are only finitely many rational solutions.

the infinite class of 4 and d\ = 2d\. Morex,y when (πι, η) = all other cases there

Note that in the above considerations d\ and d2 were given as fixed numbers. Quite recently Choudhry [Ch] provided an infinite class of solutions of (1.2) with arbitrary length and unbounded d\,d2. He observed that for arbitrary positive integers m,r,s with r > s, solutions of (1.2) are given by χ = mrsm,

di = r ( r m - s m ) ,

y = s{rm + (m - l ) s m } ,

d2 = s(rm -

sm).

As a generalisation in another direction Erdös [Er] conjectured in 1975 that for every rational number Λ the number of integral solutions ( x , y , m , n ) of x(x + 1) · · · (x + m — 1) = Λy(y + 1) · · · (y + τι — 1) min(m,n) > 3 ,

m > 1,

with y > x + m,

η > 1

is finite. The combination of Theorem 2.2 and Siegel's Theorem B, both to be stated in the next section, yields a finiteness statement in case τη and η are fixed. Actually Theorem 2.2 describes more accurately which triples (m, τι, Λ) are exceptional. The combination of Theorem 2.2 and Faltings's Theorem C gives a list of triples (m, η, X) such that the number of rational solutions x, y for every other triple is finite.

2. Diophantine equations and irreducibility of polynomials Throughout the paper we shall mean irreducible over the field of complex numbers if we merely write irreducible. The relation between irreducibility of polynomials f { X ) — g(Y) ( f ( X ) , g ( X ) G Z[X]) and solvability of diophantine equations f(x) = g(y) becomes clear from the following fundamental results. T h e o r e m Β (Siegel). The number of integral points on an irreducible algebraic curve of genus > 0 is finite. Actually Siegel gave a more refined condition which we do not state here, but refer to [Si]. In [Fa] we have the following celebrated theorem. T h e o r e m C (Faltings). The number of rational points on an irreducible algebraic curve of genus > 1 is finite.

14

F. Beukers, T.N. Shorey and R. Tijdeman

Both results are ineffective. An effective result on the finiteness of the number of integral solutions of f(x,y) = 0 if the polynomial f ( X , Y ) is irreducible over the field of rational numbers, was given by Runge [Ru], cf. [Sk, pp. 89-91]. So basic questions are to decide if a polynomial f ( X ) — g(Y) is irreducible and to compute its genus. In 1958 Ehrenfeucht [Eh] proved that f ( X ) — g{Y) is irreducible if the degrees of / and g are coprime. Three years later Davenport, Lewis and Schinzel [DLS] gave two classes of reducible polynomials f ( X ) - g(Y), namely 1.

f ( X ) = F(/xpO), g(Y) = F(9l(Y)) where F , f u g i are arbitrary polynomials, subject to deg F > 0, in which case f i ( X ) — g\{Y) is a factor of f(X)-g(Y), 2. f ( X ) = cFk(fi(X)), g(Y) = -cFk(gi(Y)) where c is a constant, / j , gi are arbitrary polynomials, k is an even integer > 2 and Fk is defined by Fk( coshö) = cosh k0. In this case u2-2uvcos f s i n 2 f is a factor of ifc(/i (X))+Fk{gi(Y)) where u = f i ( X ) , ν = gi(Y)· They further provided a criterion on the discriminants D{Λ) = disc(/(x) + Λ) and E{\) = disc(g(y) + λ) which implies that f ( X ) - g(Y) is irreducible and has positive genus and applied their criterion to the case f ( X ) = m

X

+

m-1

X

+

+

(Y}

g

=

γη

+

γη-1

+

+

γ

J

n

t h i g

w a y

t h e y

h

ed

s OW

that for these / and g and for integers η > m > 1 the equation f ( x ) = g(y) has only finitely many integral solutions. They further showed that Runge's theorem could be used to compute upper bounds for the solutions in terms of the degrees m and η of / and g, respectively, if they are not coprime. Further Nesterenko and Shorey [NeS] showed that the preceding upper bounds for the solutions can be computed in terms of m / g c d ( m , n ) and n/gcd(m, n) if gcd(m, n) > 1. Quite recently Brindza and Pinter [BrP] applied the criterion of Davenport, Lewis and Schinzel to f ( X ) = X(X

+ d1)---(x

+ (m-

l)di),

g(Y) = Y(Y + d2) • • • (Υ + (η - 1 )d2),

but they only obtained a conditional result, which is satisfied if m and η are prime or less than 31. Schinzel [Sehl] extended Ehrenfeucht's theorem in the following way. Suppose f(X),g(X) £ Q[X] and / is of prime degree. Then f(X)—g(Y) is reducible over the complex field if and only if g(Y) = f(h(Y)) and either h(X) € Q[X] or f(X)-g(Y) is of the form a(X + b)p + c(£(X))p where a,b,ceQ and 1{X) € Q[X]. The paper contains examples of the latter case with ρ = 7 and 11 due to Birch, Cassels and Guy, cf. [Ca]. Furthermore, Schinzel [Sch2] obtained the following improvement of the theorems of Runge [Ru] and Siegel [Si]. If f ( X , Y) € Z[X, Y] is irreducible over the rationale and the equation / ( x , y) = 0 has an infinity of integer solutions, then the highest homogeneous part of f ( X , Y) is, up to a constant factor, a power of a linear or an irreducible indefinite quadratic form. In his book, published in 1982, Schinzel [Sch3, Section 8, Theorem 11] gave another classification of polynomials f(X),g(X) € Z[X] such that f { X ) - g(Y) is reducible. Fried made a deep study of the structure of the factors of f ( X ) — g(Y)· We mention some special cases of his results which are relevant to our work. For

Irreducibility of polynomials and arithmetic progressions

15

a survey we refer to Fried's own contribution to these Proceedings. Pried and MacRae [FrM] showed that fi{X) - gi(Y) divides f ( X ) - g(Y) in C[X, Y] if and only if there exists a polynomial F(T) G C[T] such that f ( X ) = F ( / i ( X ) ) and g(Y) = F(gi(Y)). This is due to Schinzel in case deg(/) is prime. The decomposition properties were elaborated in [Frl]. Pried proved (cf. [Frl, Propositions 2 and 3]) that if f ( X ) , g{X) G Z[X], then there exist polynomials fi,f2,9i,92 € Z[X] such that 1. 2. 3. 4.

f(X) = Mf2(x)),g(Y)=9i(92(Y)), deg(A) = deg(gi), the splitting fields of fi{X) — X and gi(X) — X are the same, the irreducible factors of f ( X ) — g(Y) are in one-to-one correspondence with the irreducible factors of f i ( X ) - 9i(Y)·

In Proposition 1 of [Fr2] Pried used the Riemann-Hurwitz formula to give expressions for the genus of f ( X ) — g(Y)· Fried used the formulas to describe the structure of polynomials / , g € K[X], Κ a number field, such that f ( X ) — g(Y) has an irreducible factor which defines a curve having infinitely many /^-integral points (cf. Corollary of Theorem 3 of [Fr2]). By using Schinzel's characterization of reducible polynomials f ( X ) — g(Y) [Sch3] and by using Fried's decomposition theorem on such polynomials it is possible to derive the following result. Theorem 2.1. Let m and η be positive integers with m < η and let Λ e C*. If X(X + 1) · · · (X + m - 1) - XY(Y + 1) · · · (Υ + η - 1) is reducible in C[X, Y] then one of the following possibilities holds: 1. 2. 3.

τη = η, Λ = 1, in which case X — Y is a factor, m = η is odd, Λ = —1, in which case X + Υ + m — 1 is a factor, m = 2, η = 4, λ = in which case we have 4X{X + 1) - Y{Y + 1)(F + 2)(Y + 3) = {2X - Y

2

- 3Y)(2X + 2 + 3Y + Y2).

In the next section we shall give a selfcontained proof of this theorem. The following genus computation allows us to apply Theorems Β and C. Theorem 2.2. Consider the curve X{X + 1) · • · (X + m - 1) = \Y(Y

+ 1) · · · (Υ + η - 1)

with η > m > 1 and Λ e C*. Suppose it is irreducible. Then its genus is zero in the following cases: 1. m = 2, η = 2, 2. m = 2, η = 3, Λ = ±3-^/3/8, 3. τη = 2, η = 4, λ = -4/9, 4. πι = 2, η = 6, λ = ( - 1 0 ± 7\/7)/576. The genus is one in the following cases: 1. 2.

m = 2, η = 3, λ φ ±3\/3/8, τη = 2, η = 4, Λ φ - 4 / 9 ,

16 3. 4. 5. 6. 7. 8.

F. Beukers, Τ.Ν. Shorey and R. Tijdeman

τη τη τη τη τη τη

= 2, η = 5, Λ = —l/4t, 3125t 4 - 47500ί 2 + 82944 = 0, = 2, η = 6, λ = 16/225, = 2, η = 8, λ = - 1 / 4 1 , t3 + 56712 - 544321 - 4665600 = 0, — 3, η — 3, = 3, η = 4, Λ = ± 3 \ / 3 / 2 , = η = 4, λ = - 9 / 1 6 , - 1 6 / 9 .

In all other cases the genus is strictly bigger than one.

3. Proof of Theorem 2.1 Let / € C[X] and let Sj, Sg be the set of stationary points of / , g which we assume to be simple. Let m = deg(/) and η = deg(g). For any a e C let ma = #{« e Sf I f(a) = a} na = #{/? € Sg I g(ß) = a} Consider the polynomial f ( X ) — g{Y) and suppose it is reducible. P r o p o s i t i o n 3.1. Suppose m = η and f ( X ) - g(Y) = F(X,Y)G(X,Y) deg(i 7 ') = m i , deg(G) = ni2 with πΐ\,πΐ2 > 1 and πΐ\ + τη-χ — m. Then

with

πΐ\πΐ2 < y ^ m 0 n a . aec Proof. Geometrically the curve C given by f ( X ) — g{Y) = 0 consists of two components C i , C 2 given by F(X,Y) — 0 and G(X, Y) = 0. By homogenisation of the coordinates we can assume that C is embedded in projective space P 2 . Let f ( X ) = Σ™=ο fTXr and g(Y) = Σ™=ο grYr• The points of intersection of C with the line at infinity are given by fmXm — gmYm· These are distinct points with multiplicity one. So C has no singularities at infinity. The finite singular points of C are given by the pairs (a, ß) such that f'(a) = g'(ß) — 0 and f(a) = g(ß). The local equation of C around such a point looks like 0 = / » ( X - a ) 2 - g"{ß){Y

- ßf

+ ...

Since / " ( α ) , g"(ß) / Owe conclude that the singularities are simple. In total there are ^ a 6 C mana singular points. Any point of intersection between F = 0 and G = 0 is a singular point of C. Since all singularities are simple, the order of intersection is one. By Bezout's theorem there are 77117712 points of intersection, hence τηχπΐ2 < S a e C 1 7 1 a n a• We next consider the case of unequal degrees m, n. Let us introduce the weighted degree δ given by S(XaYb) =na + mb and notice that δ [AB) = δ(Α) + δ (Β) for all Α,Β £ C[X,Y], AB φ 0. For any A £ C[X,Y] we denote the highest degree part with respect to δ by (A)h- We also have ( A B ) h = (A)h(B)h· Assume that f(X)-g(Y) =

F(X,Y)G(X,Y)

Irreducibility of polynomials and arithmetic progressions

17

where F, G have positive degree. Prom comparison of the highest degree parts we get fmXm - gnYn = (F)h(G)h. So (F)h must be of the form aXr +... + bYs with a, b φ 0. The monomials Xr, Ys have equal weighted degree, i.e. nr = ms and so, r, s must be multiples of m/d, n/d respectively, where d = gcd(m, n). In particular the weighted degrees of F and G are multiples of mn/d and also d > 2. Note, by the way, that this immediately implies Ehrenfeucht's theorem. Proposition 3.2. Let the notation be as above and let τηι,τη^ be the weighted degrees of F, G respectively. Then, τη\πΐ2 < mn^^ mana. aec Moreover, mj,m2

are multiples of mn/d and mi + m? = mn.

Proof. Choose be C such that b -l)=d(m/d-l)+ £ i aeSf = τη — d+

£ 1 y0:g(yo)=f(a)

^^ (n — 2ra) aeSf

Hence 2gc = 2 - 2m + ^ ( e * - 1) = ] T (ra - 2 r a ) - m + 2 - gcd(ra, n). i a£Sf

•

Irreducibility of polynomials and arithmetic progressions

21

Proof of Theorem 2.2. We shall apply Proposition 4.1 to the curve f ( X ) - g(Y) = 0 with f { X ) = X(X + 1) · · · (X + m - 1) and g(Y) = XY(Y + 1) χ . . . χ (Y + m— 1). We may assume that 1 < m < n. By Proposition 3.4 we know that ra < 2 in all cases and ra < 1 if η is odd. Write S(n) = 2 when η is odd and δ(η) = 4 when η is even. Since |5/| = πι — 1, we get 2gc = ^^ (η — 2 r a ) — τη + 2 — gcd(m, n) aesf > (η — S(n))(m — 1) + 2(1 — to) + m — gcd(m, n) = (η — δ(η) — 2 )(m — 1) + m — gcd(m, η) Suppose η > 9 or η = 7. Then n— 3 and we get 2gc > 3(πι— 1), hence gc > 1· Suppose η = 8, then η-ό(η)-2 = 2 and 25c > 2(m—l)+m —gcd(m, 8) > 2(m — 1). Hence gc > 1 if m > 3. This leaves us the case m = 2, η = 8. Suppose η = 5. Then 2gc > m — 1 + m — gcd(m, 5). We easily check that for all choices of m = 3,4,5 we get gc > 1· This leaves us with m = 2, η — 5. We must now determine the genera in the following remaining cases: 1. m 2. πι 3. m 4. m 5. to

= 2, η = 2,3,4,5,6,8 — 3, η — 3,4,6 = 4, η = 4,6 = 5, η = 6 = 6, η = 6.

To get gc for these cases we must compute the set of stationary values of each of the polynomials hm — X ( X + 1) · · · (Χ + πι— 1) with m = 2,3,4,5,6,8. In other words, we compute {hm(a) \ h'm(a) = 0} for m = 2 , 3 , . . . , 8. πι = 2

{ - 1 / 4 } (once)

πι = 3 πι = 4

{zeros of 27χ2 — 4} (once)

πι = 5 πι = 6 πι = 8

{zeros of 3125a;4 - 47500a:2 + 82944 (once)}

{9/16 (once),—1 (twice)} {-225/64 (once) and zeros of 27a;2 + 320a; - 2304 (twice)} {11025/256 (once) and zeros of x 3 + 567a;2 - 54432a; - 4665600 (twice)}.

The adjective 'twice' indicates that there are two stationary points above the stationary value involved. Let TO = 2. Then Sf consists of a single point and f ( S f ) = —1/4. If g(Sg) does not contain this point then 2gc = η — m + 2 — gcd(m,η) = η — gcd(n, 2). Hence gc > 1 if η — 5,6,8, gc = 1 if η = 3,4 and gc = 0 if η = 2. If - 1 / 4 is an element of g(Sg) then the value Λ is determined by this and we can easily compute the genus. The result is in the statement of our theorem. Note that m = 2, n = 4, λ = 1/4 is excluded since it is reducible. Let to = 3. If f ( S f ) is disjoint with g(Sg) we get 2 gc = 2 n - m + 2 - g c d ( m , n) — 2n - 1 - gcd(n, 3). Hence gc > 1 if η = 4,6 and gc = 1 if η = 3. If f ( S f ) Π g(Sg)

22

F. Beukers, Τ.Ν. Shorey and R. Tijdeman

is non-empty we get explicit values for Λ which are enumerated in the theorem. The cases m = n = 3,A = ± 1 are excluded because they are reducible. When m = 4,5,6 we argue in exactly the same way as above. •

5. P r o o f of Theorem 1.1 Proposition 5.1. Let I e Q*. Then each of the following genus one curves 1. 2. 3. 4.

y(y + 1) = lx(x + l)(z + 2) y{y + 1 ){y + 2) = l3x(x + l)(z + 2) with l φ ± 1 y(y+ 1) = l2x{x+ l)(z + 2)(ζ + 3) 225y(y + 1) = 16a;(ar + l)(ar + 2) (a; + 3)(x + 4)(x + 5)

contains infinitely many rational

points.

Proof. We use a deep result of B. Mazur [M] which says that the rational torsion group of an elliptic curve over Q is either Z/7VZ with TV = 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 1 0 , 1 2 or Z/2Z χ Z/2./VZ with Ν = 1,2,3,4. In particular, if an elliptic curve contains finitely many points, this number is at most 16. The number of torsion points whose orders do not divide 2 is at most 12. Let us apply this principle to the first curve y(y + 1) = lx(x + l)(a; + 2). There we have the six obvious rational points (a, b) with a = 0, — 1 and 6 = 0, —1, —2. By using the cord method for elliptic curves we find the additional points

( - > + π ) · ( ί . - ' - 7 Μ τ · > + ί)· (Ζ-1,Ζ2-1),α-1,-/2).

A straightforward check, long and boring by hand, but swiftly done using a computer algebra system, reveals that if l φ ± 1 / 4 , ± 1 / 2 , ± 1 , ± 3 / 4 , ± 2 / 3 then none of the rational points coincide or coincide with a 2-torsion point. Note that 2-torsion points can be recognized by the fact that their (/-coordinate is —1/2 in our curve. Hence we have found more than 12 rational points whose order does not divide 2 and so our curve must have infinitely many rational points. The remaining cases can be checked simply by finding at least 13 points whose orders do not divide 2. We exhibit our finds here. For I = 1 we found (0,0), (0, - 1 ) , ( - 1 , 0 ) , ( - 1 , - 1 ) , ( - 2 , 0 ) , ( - 2 , - 1 ) , (1,2), (1, - 3 ) , (5,14), (5, - 1 5 ) , (-3/4, -3/8), (-3/4, -5/8), (-14/9, -35/27), (-14/9,8/27). For I — 1/4 we found (0,0), ( 0 , - 1 ) , ( - 1 , 0 ) , ( - 1 , - 1 ) , ( - 2 , 0 ) , ( - 2 , - 1 ) , (2, - 3 ) , (4, - 6 ) , (4,5), (2,2), (-20/9, -22/27), (-20/9, -5/27), ( - 3 / 4 , -15/16), (-3/4, -1/16).

Irreducibility of polynomials and arithmetic progressions

23

For I — 1/2 we found (0,0), (0, - 1 ) , ( - 1 , 0 ) , ( - 1 , - 1 ) , ( - 2 , 0 ) , ( - 2 , - 1 ) , (2,3), (2, - 4 ) , (3,5), (3, - 6 ) , ( - 1 / 2 , - 3 / 4 ) , ( - 1 / 2 , -1/4), ( - 4 / 9 , -20/27), ( - 4 / 9 , -7/27). For I = 3/4 we found (0,0), (0, - 1 ) , ( - 1 , 0 ) , ( - 1 , - 1 ) , ( - 2 , 0 ) , ( - 2 , - 1 ) , (4,9), (4, - 1 0 ) , (4/3, -10/3), (4/3,7/3), ( - 1 / 4 , -7/16), ( - 1 / 4 , -9/16), ( - 2 / 3 , -1/3), ( - 2 / 3 , -2/3). For I = 2/3 we found (0,0), (0, - 1 ) , (-1,0), ( - 1 , - 1 ) , (-2,0), ( - 2 , - 1 ) , ( - 1 / 3 , -4/9), ( - 1 / 3 , -5/9), (11/3, -77/9), (11/3,68/9), ( - 5 / 8 , -11/16), ( - 5 / 8 , -5/16), (3/2,5/2), (3/2, -7/2). For negative I we can remark that the curves with I and —I are isomorphic via (x,y) ( - 3 - x,y). Consider the curve y(y + 1 )(y + 2) = l3x(x + l ) ( z + 2) with l φ ±1. The point ( - 1 , - 1 ) is a point of inflexion and can be used as the zero of our addition law. It turns out that the order 2 points all lie on the line at infinity. Obvious rational points on the curve are given by (a, b) with a, 6 € { 0 , - 1 , - 2 } . Further rational points are given by / / -3 Vz3

+

12

Ζ3 + 4\ P^sJ'

—3/3 i'z3

+ r'

3 (

/

l3 + 2\

z -i'/ -i/' 3

3

2Z3 ± 1 —12Z3 \ δΡΤϊ'δΡΤϊ; (

2/3 — 1

2Z3 - 1\

^

sz -i'

sz3 - 1 y '

3

So when l φ ± 2 , ±1/2 we have found 13 finite points whose orders do not divide 2. Again, a long and boring check reveals that there are no rational values of I for which two of these points coincide. For I = 2 vre found the points (0,0), (0, - 1 ) , (0, - 2 ) , (-1,0), ( - 1 , - 2 ) , (-2,0), ( - 2 , - 1 ) , ( - 2 , - 2 ) , (-1/3, -8/3), (-5/3,2/3), (3/7,10/7), (-17/7, -24/7), (-20/21, -10/21). The curves with I = —2, ±1/2 are trivially isomorphic to the one with I = 2. So in all cases we find at least 13 finite rational points beside ( - 1 , - 1 ) . Hence our proposition is proved again in this case. Now consider the curve y(y + 1) = l2x(x + l)(:r + 2)(x + 3). This curve has two involutions given by y —• — 1 — y and χ —3 — x. They generate a group of order 4. At infinity we have two rational points. Consider the four points (n rrt (-Λ frt ( - Z J — 1 (* + l ) ( 3 * - l h ( 2/2 + 3f + 1 4f 4 - Ρ + 2Z - 1 χ lu,uM , 41 Μ 21 Al Γ 2l The only rational values of I for which any such point can lie in the orbit of another point are given by I = ± 1 , ±1/3, ±1/2. In all other cases we have now, after application of the group action, 16 rational points on the curve plus two points at infinity. Hence there are infinitely many rational points. In the exceptional cases

24

F. Beukers, Τ.Ν. Shorey and R. Tijdeman

I = ± 1 , ±1/3 we exhibit at least 16 finite rational points. For I = ± 1 we find (0,0), (0, - 1 ) , ( - 1 , 0 ) , ( - 1 , - 1 ) , ( - 2 , 0 ) , ( - 2 , - 1 ) , ( - 3 , 0 ) , ( - 3 , - 1 ) , (4/3, -65/9), (4/3,56/9), (-13/3,56/9), (-13/3, -65/9), (-5/4,5/16), ( - 5 / 4 , -21/16), ( - 7 / 4 , -21/16), (-7/4,5/16). For I = ±1/3 we find (0,0), (0, - 1 ) , ( - 1 , 0 ) , ( - 1 , - 1 ) , ( - 2 , 0 ) , ( - 2 , - 1 ) , ( - 3 , 0 ) , ( - 3 , - 1 ) , ( - 9 / 4 , -15/16), ( - 9 / 4 , -1/16), ( - 3 / 4 , -1/16), (-3/4, -15/16), (1/3,8/27), (1/3, -35/27), (-10/3, -35/27), (-10/3,8/27). The case I = ±1/2 corresponds to a reducible curve which has infinitely many rational points in a trivial way. Finally consider the genus one curve 225y(y + 1) = 16x(x + l ) ( x + 2){x + 3) χ (χ + 4) (χ + 5). There are two rational points at infinity and we have the finite rational points (a, b) with a = 0, —1, —2, —3, —4, —5, b = 0, — 1 and (-43/5,33024/625), (9/5,9728/625),

(-43/5, -33649/625), (9/5, -10353/625).

Hence we have at least 18 rational points so there must be infinitely many.

•

Proof of Theorem 1.1. Let C be the plane algebraic curve given by

X{X + DI) · · · (X + (TO - L)DI) = Y(Y + d2) • • • (Υ + (η - 1 )d2). After the replacements X —> d\X, Y —» d2Y we see that C is isomorphic to

X(X + 1) · · · (X + m - 1) = ΛY(Y + 1) · · · (Υ + η - 1) with Λ = d2/d™. We know that this curve is irreducible unless m = η, A = ± l or TO = 2, η — 4, λ = 1/4. The first case does not occur since Λ φ ± 1 . The latter case gives rise to the factorisation

X(X + 2d2) - Y(Y + d)(Y + 2d)(Y + 3d) = (X-Y2-

3dY)(X + Y2 + MY + 2d2)

which induces the infinite set of integral solutions given in our theorem. In all other cases C is irreducible. From our Theorem 2.1 we know that the genus of C can only be zero if TO = η = 2. In that case our equation can be rewritten as {2x + d\)2 — (2y + d2)2 = d2 — d2 which has clearly only finitely many solutions. In all other cases the genus of C is positive and by Siegel's Theorem Β the number of integral points is finite. This proves the first part of our theorem. We now consider rational solutions. When πι = η = 2 there are clearly infinitely many rational solutions since we have a conic with at least one rational point on it. By Theorem 2.1 there are no other cases of genus zero. We now proceed to the genus one cases. According to Theorem 2.1 these are given by (TO, n) = (2,3), (2,4), (3,3) and (τη,η) = (2,6) with λ = 16/225. According to Proposition 5.1 we have infinitely many rational solutions in all cases. This gives

Irreducibility of polynomials and arithmetic progressions

25

rise t o the second assertion of our theorem. Finally, according to Theorem 2.1 the remaining cases have all genus at least two and hence, by Faltings's Theorem C, we have finitely many rational solutions. • A c k n o w l e d g e m e n t . We thank B. Brindza, M.D. Fried, M. Nori and A. Schinzel for their valuable remarks on the background of the solutions of our problems. T h e y alerted us t o most of the papers mentioned in Section 2. T h e paper would not have had its present form without the lively conference in wet Zakopane.

References [BoK]

Boyd, D.W., Kisilevsky, H.H., The diophantine equation u ( t t + l ) ( u + 2 ) ( w + 3 ) = v(v + l)(v + 2). Pacific J. Math. 40 (1972), 23-32.

[BrP]

Brindza, B., Pinter, Α., On the irreducibility of some polynomials in two variables. Preprint.

[Ca]

Cassels, J.W.S., Factorization of polynomials in several variables. In: Proceedings of the Fifteenth Scandinavian Congress (Oslo, 1968) (Lecture Notes in Math. 118), 1-17. Springer, Berlin 1970.

[Ch]

Choudhry, Α., On arithmetic progressions of equal lengths and equal products of terms. Acta Arith. 82 (1997), 95-97.

[DLS]

Davenport, H., Lewis, D.J., Schinzel, Α., Equations of the form f(x) Quart. J. Math. Oxford Ser. (2) 12 (1961), 304-312.

[Eh]

Ehrenfeucht, Α., A criterion of indecomposability of polynomials (in Polish). Prace Mat. 2 (1958), 167-169.

[Er]

Erdös, P., Problems and results on number theoretic properties of consecutive integers and related questions. In: Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Congressus Numerantium No. XVI), 25-44. Utilitas Math., Winnipeg 1976.

[Fa]

Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73 (1983), 349-366.

[Frl]

Fried, M., The field of definition of function fields and a problem in the reducibility of polynomials in two variables. Illinois J. Math. 17 (1973), 128-146.

[Fr2]

— On a theorem of Ritt and related diophantine problems. J. Reine Angew. Math. 264 (1973), 40-55.

[Fr3]

— Irreducibility results for separated variables equations. J. Pure Appl. Algebra 48 (1987), 9-22.

[FrM]

Fried, M., MacRae, R.E., On curves with separated variables. Math. Ann. 180 (1969), 220-226.

[Ga]

Gabovich, Ya., On arithmetic progressions with equal products of terms (in Russian). Colloq. Math. 15 (1966), 45-48.

[GrH]

Griffiths, P., Harris, J., Principles of algebraic geometry. Wiley, New York 1978.

[MaB]

MacLeod, R.A., Barrodale, I., On equal products of consecutive integers. Canad. Math. Bull. 13 (1970), 255-259.

=

g{y).

26

F. Beukers, Τ.Ν. Shorey and R. Tijdeman

[MeJ

M^kowski, Α., Problemes P543 et P545, RL. Colloq. Math. 19 (1968), 179-180.

[M]

Mazur, B., Modular curves and the Eisenstein ideal. Inst. Hautes Etudes Sei. Publ. Math. 47 (1977), 33-186.

[MiS]

Mignotte, M., Shorey, T.N., The equations ( x + 1 ) · · · (x+fc) = (y+1) · • · ( y + m k ) with m = 5,6. Indag. Math. (N.S.) 7 (1996), 215-225.

[Mo]

Mordell, L.J., On the integer solutions of y(y + 1) = x(x + l ) ( x + 2). Pacific J. Math. 13 (1963), 1347-1351.

[NeS]

Nesterenko, Yu.V., Shorey, T.N., An equation of Goormaghtigh. Acta Arith. 83 (1998), 381-389.

[Ru]

Runge, C., Uber ganzzahlige Lösungen von Gleichungen Veränderlichen. J. Reine Angew. Math. 100 (1887), 425-435.

[SaSl]

Saradha, N., Shorey, T.N., On the ratio of two blocks of consecutive integers. Proc. Indian Acad. Sei. Math. Sei. 100 (1990), 107-132.

[SaS2]

— The equations (x + 1) • · · (a; + k) = (y + 1) · · · (y + mk) with m — 3,4. Indag. Math. (N.S.) 2 (1991), 489-510.

[SaS3]

— On the equation x(x + di) • • • (x + (k — l ) d i ) = y(j/ + g

+

|a,|,

where log + ν = max(logi>, 0), for υ > 0, and log + 0 = 0. For polynomials with integer coefficients, clearly m(P(x)) > 0 with the equality only if P(x) is monic and has all its zeros inside the unit circle, and hence is a product of a monomial xa and a cyclotomic polynomial, by Kronecker's theorem. In [Le], Lehmer noted that m(P(x)) measures the growth rate of the sequence An = Π?=ι(α" — 1)' a n d asked whether m(P) can be arbitrarily small but positive for P(x) e Z[xj. The smallest value he was able to find was m(x10 + χ9 - χ7 - χ6 - χ5 - χ4 - χ3 + χ + 1) = log(l.17628081...) = .16235761

(1)

As we pointed out in [Bol], Lehmer's question leads in a natural way to the consideration of m(P(xi,..., xn)) since one has lim τη(Ρ(χ,χη))

=

m(P(x,y)).

This formula, and its generalization to η variables by Lawton [La] show that measures of polynomials in many variables are limit points of measures of polynomials in one variable. We conjectured in [Bol] that the set L of all measures m(P(xi,... ,xn)) for polynomials with integer coefficients should be a closed set. This would imply a positive answer to Lehmer's question. Smyth [Sm] showed that the values of m(P) could be interesting even for very simple polynomials by proving that m(l + x + y) = ^ L ( x _ 3 , 2 ) = L'(X_3,

-1)

(2)

where the second form follows from the functional equation for L'(\_3,s). It is interesting that (1) can also be written in a similar way. Lehmer's polynomial P(x) = ζ 1 0 + χ9 — χ7 — χ6 — χ5 — χ4 — χ3 + χ + 1 is the minimal polynomial of a Salem number (a real algebraic integer σ > 1 for which l / σ is a Galois conjugate and all other conjugates lie on the unit circle). Thus m(P(x)) = logσ. Chinburg [Ch], using a result of Stark [St], showed that for every Salem number σ, if Κ = Q(CT) and k = Q(a + l/er), if χ is the nontrivial character of G&\(K/k) and if L(\, s) is the Artin L-function of χ, then logCT =

rL'(x,0),

where r is an explicit rational number (in which the class numbers h(K) and h(k) appear).

Mahler's measure and special values of L-functions — some conjectures

29

Perhaps motivated by this, and certainly motivated by Smyth's result (2), Chinburg was led to ask, for each real odd Dirichlet character X_p whether there is a polynomial P f ( x , y) with integer coefficients such that m ( P f ( x , y)) is a rational multiple of L ' ( x _ j , —1), i.e. does Smyth's formula generalize. His student, Ray [Ra] constructed such polynomials for the six values 3, 4, 7, 8, 20 and 24 using the theory of dilogarithms. In fact, his proof for / = 7 requires the proof of a new multivariable dilogarithm identity and gives: l) 2

πι(Φ7(χ)(ν -

+ 7x

2

(x +

1)

2

y)

= ®Ζ/(χ_ 7 , - 1 ) ,

(3)

where Φγ(χ) = (χ 7 — 1)/(χ — 1) is the 7th order cyclotomic polynomial. Recently, we have found rather simpler formulas than Ray's and extended the list of conductors by adding 11,15, 35, 39, 55 and 84. However, most of our formulas have only been verified numerically to 50 decimal places rather than being proved. For example, in contrast to (3), we have m{(x

+ l)

2

y+ (χ2+ χ +

and m ( ( x

2

1 1 = - d

1))

+ χ + l ) y + ( x 2 + 1))

,

7

? 1

= — d

1 5

(4)

,

(5)

and m({x

+ l)

2

{ x 2 + x + l )y + ( x

2

- x+ l)2) = \ d

Ο

u

(6)

.

Our notation here is that j-3/2 d

f

= L ' ( x _

p

- l ) =

J

— L ( x _

f

, 2 ) ,

(7)

?

and the symbol =, read "conjectured to be equal to" means that the two members of the equation are equal to many decimal places (usually 50). Our examples are of the form m ( A ( x ) y + B ( x ) ) where A and Β are cyclotomic polynomials, so that m { A { x ) y+ B { x ) ) = m

+

( B ( x ) / A ( x ) ) ,

where m

+

( P ) =

[ Jo

•••

[ Jo

log+|P(e(ii),---,e(in))l24 comes from the fact that the Jacobian of the curve P(x, y) = 0 is an elliptic curve of conductor 24, while the term d3 comes from the particular way P(x,y) = 0 intersects T 2 . Of course, it would be easy to construct formulas such as (12) with reducible polynomials P(x,y), but here P(x,y) is irreducible. Among the examples we discovered in our early experiments is the following example: m((x2

+x + 1 )y2 + (x2 + x)y + ( z 3 + x2 + x)) = (1/3)634·

(13)

This is at first no more or less surprising than (9), (10) or (11), until one realizes that the curve P(x,y) — 0 has genus 2. The Jacobian of this curve happens to split into the product of two elliptic curves, of conductors 34 and 17 and m{P) picks out one of these two factors.

32

David W . Boyd

3. W h a t is known? The paper [Bo2] describes an attempt to understand and generalize such formulas by means of numerical experiment. The idea is to study certain one-parameter families of polynomials which define curves of genus 1 and 2 in order to determine conditions on the polynomials P(x,y) so that formulas such as (9) (11), and (13) should hold. The example m(y2 — x3 — k) = m(x + y + k) = log |/c|,

if |fc| > 2,

shows that it is not sufficient that P(x,y) = 0 define an elliptic curve nor is it necessary, as example (13) shows. For example, generalizing (10), we found that m((x + 1 )y2 + (χ2 + kx + 1 )y + x{x + 1)) = rkL'(Ek,

0),

for all integer \k\ < 100 for which the polynomial defines an elliptic curve Ek, and where rk is a rational number which seems usually to be the reciprocal of an integer. In degenerate cases, i.e. for values of k for which the curve P(x, y) = 0 is rational, the measure seems to be a multiple of an appropriate Ζ/(χ, —1). Similarly, generalizing (11), we find that m(y2 + (x2 +kx-

1 )y + χ3) = rkL'(Ek,

0),

for integers k with 2 < |fc| < 20. The reason for the omission of the integers —1, 0, 1 is explained in [Bo2]. For these values of k the condition (B), mentioned below, is not satisfied. One condition (A) that appears to be necessary, is that the "faces" Pp of the polynomial P(x,y) must have Mahler measure 0. Here the faces are defined in terms of the Newton polygon N(P) of Ρ which is the convex hull in R 2 of the set of lattice points (i,j) for which the monomial xlyJ appears as a term in P(x,y). A face F of N(P) is the intersection of N(P) with a support line to N(P), and a face PF of Ρ is the sum of the monomials making up Ρ over all lattice points in F. Each PF{X, y) is essentially a polynomial in one variable and the condition τTI(PF) = 0 means that this polynomial is cyclotomic. A second condition (B) has to do with the way in which the complex curve P{x, y) = 0 links the torus T 2 . This is described precisely in [Bo2]. We conjectured in [Bo2], for polynomials of the form P(x,y) = A(x)y2 + B(x)y + C(x), with deg(AC - AB2) < 4, that if P(x,y) = 0 has genus 1, and satisfies (A) and (B), then m(P) is a rational multiple of L'(E, 0) where Ε is the Jacobian of P(x, y) = 0. Recently, Rodriguez Villegas [RV] and independently, Hubert Bornhorn (private communication), have shown how this conjecture is related to Beilinson's conjectures. The condition (B) can be used to show that m(P) is a rational multiple of the Beilinson regulator applied to the symbol {x,y} and the condition (A) shows that some power of {x, y} is in the kernel of the tame symbol (if one accepts a conjecture concerning the rank of a certain K-group). Putting these together shows that the above conjecture is a special case of the Beilinson conjectures.

Mahler's measure and special values of L-functions — some conjectures

33

If Ε has complex multiplication, then in many cases the conjectured formulas can actually be proved due to the more explicit knowledge about L(E,s). Rodriguez Villegas [RV] has also given a very interesting interpretation of many of the formulas in terms of certain non-holomorphic modular forms. Nothing of this nature has yet been proved about the genus 2 examples constructed in [Bo2]. For the two classes of examples considered there, the polynomials Ρ all vanish on the torus so Deninger's result does not apply directly. However, it does appear that conditions (A) and (B) correctly predict the existence of formulas of the type m{P) = rL'(E, 0), so presumably there is some connection with the Beilinson conjectures. Then there are also the examples of the type (12) for which there should be a cohomological explanation. Finally, since the formulas described here are of a very explicit and concrete nature, it would be interesting to have analytic proofs. For example, the curve Ε is of (9) is modular and the normalized cusp form is easily seen to be oo

oo

f ( z ) = Σ nqn = q Π(! n=1 n=1

" 9")(1 -